Solutions of the bi-confluent Heun equation in terms of the Hermite functions

TL;DR

This paper develops a method to express solutions of the bi-confluent Heun equation using Hermite functions, explores conditions for finite-sum solutions, and applies these findings to specific quantum potentials, including an infinite well.

Contribution

It introduces a series expansion of the bi-confluent Heun equation in Hermite functions and identifies parameter restrictions for finite solutions, with applications to quantum potentials.

Findings

Derived explicit solutions for specific quantum potentials.

Identified conditions for finite-sum Hermite function expansions.

Provided exact and approximate energy spectra for bound states.

Abstract

We construct an expansion of the solutions of the bi-confluent Heun equation in terms of the Hermite functions. The series is governed by a three-term recurrence relation between successive coefficients of the expansion. We examine the restrictions that are imposed on the involved parameters in order that the series terminates thus resulting in closed-form finite-sum solutions of the bi-confluent Heun equation. A physical application of the closed-form solutions is discussed. We present the five six-parametric potentials for which the general solution of the one-dimensional Schr\"odinger equation is written in terms of the bi-confluent Heun functions and further identify a particular conditionally integrable potential for which the involved bi-confluent Heun function admits a four-term finite-sum expansion in terms of the Hermite functions. This is an infinite well defined on a half-axis. We present the explicit solution of the one-dimensional Schr\"odinger equation for this potential and discuss the bound states supported by the potential. We derive the exact equation for the energy spectrum and construct an accurate approximation for the bound-state energy levels.